Question

The equation of a progressive wave for a wire is :
$y=4\sin \left[\frac{\pi }{2}(8t-\frac{x}{8})\right]$. If $x$ and $y$ are measured in $\text{cm}$ and $t$ is in $\text{seconds},$ then velocity of wave is :

• A. $64\text{cm/s}$ along $-x$ direction
• B. $32\text{cm/s}$ along $-x$ direction
• C. $32\text{cm/s}$ along $+x$ direction
• D. $64\text{cm/s}$ along $+x$ direction

Answer 1

The correct option is D $64\text{cm/s}$ along $+x$ direction

$y=4\sin \left[\frac{\pi }{2}(8t-\frac{x}{8})\right]=4\sin (4\pi t-\frac{\pi }{16}x)$
Compare above equation by $y=A\sin (\omega t-kx)$, we get
$\omega =4\pi$ ; $k=\frac{\pi }{16}$
from formula
wave speed $\left(v\right)=\frac{\omega }{k}=64\text{cm/s}$
Direction of wave is in $+x$-direction because $y=A\sin (\omega t-kx)$ is the equation of wave travelling along $+x$-direction.
Hence, option $\left(d\right)$ is the correct answer.